Comprehend how the scale factor impacts side lengths, perimeters and areas of similar shapes with this gamut of scale factor worksheets. Also, learn to find the ratio of areas and perimeters of similar polygons. The worksheets are available in both customary and metric units. The pdf worksheets are recommended for 7th grade, 8th grade, and high school. Explore some of these worksheets for free!
Select the Measurement Units
Relation between Scale Factors and Ratios
Compare the similar figures and answer the series of questions that follow. Find the area, perimeter and scale factor, understand the influence of scale factor on ratio of areas and perimeters as well.
Area and Perimeter - Table Form
Packed in these printable ratio and scale factor of area and perimeter worksheets for grade 7 are tables with missing parameters. Complete them by determining the ratio and perimeter in Table A, ratio and area in Table B; based on the provided scale factor.
Ratios - Area and Perimeter
Reiterate the concept of scale factor by applying it to solve word problems. Compute the area and perimeter of similar triangles and similar polygons.
Perimeter of Similar Figures | Level 1
Reiterate the fact that the ratio of the perimeters of similar figures is the same as the scale factor, and figure out the missing perimeter in these level 1 worksheets.
Perimeter of Similar Figures | Level 2
Level up with these printable level 2 worksheets! Equate the ratio of side lengths with the apt ratio of the perimeters, apply the cross product property, and solve for the unknown perimeter.
Area of Similar Figures | Level 1
Specifically dealing with determining the area of similar shapes using the given scale factor, these level 1 problems can be easily solved by applying the area of similar polygons theorem.
Area of Similar Figures | Level 2
Obtain the scale factor using the given side lengths; equate the ratio of the areas with the square of this scale factor, make the unknown area the subject, and solve.
Perimeter and Area of Similar Figures | Level 1
Practice these grade 8 and high school pdfs that consist of simple word problems to find the area or perimeter of the original or dilated image and grasp the impact of scale factor on the ratios of area and perimeter.
Perimeter and Area of Similar Figures | Level 2
These word problems feature similar special quadrilaterals and polygons with up to 10 sides. Recall that the square of the ratio of perimeters equals the ratio of the areas, and solve for the unknown value.
Missing Sides of Similar Figures
Determine the missing sides of similar triangles or similar polygons using the information provided in these printable worksheets. Few problems are offered in the word format.
If you're seeing this message, it means we're having trouble loading external resources on our website.
If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.
In Chapter 6 you learned that if two polygons are similar, then the ratio of their perimeters, or of any two corresponding lengths, is equal to the ratio of their corresponding side lengths. As shown below, the areas have a different ratio. Ratio of perimeters
Blue Red
Ratio of areas
10t 10
Blue Red
}5}5t
6t 2 6
}5}5t
2
3t
3 2
2t
For Your Notebook
THEOREM THEOREM 11.7 Areas of Similar Polygons
If two polygons are similar with the lengths of corresponding sides in the ratio of a : b, then the ratio of their areas is a 2 : b 2. Side length of Polygon I Side length of Polygon II
a b
b
}}} 5 }
Area of Polygon I Area of Polygon II
a
a2 b
Justification: Ex. 30, p. 742
EXAMPLE 1
II
I
}} 5 }2
Polygon I , Polygon II
Find ratios of similar polygons
In the diagram, n ABC , n DEF. Find the indicated ratio. a. Ratio (red to blue) of the perimeters INTERPRET RATIOS You can also compare the measures with fractions. The perimeter of n ABC is two thirds of the perimeter of nDEF. The area of n ABC is four ninths of the area of nDEF.
D
F B
b. Ratio (red to blue) of the areas
8
12 C
A
E
Solution 8 2 5} , or 2 : 3. The ratio of the lengths of corresponding sides is } 12
3
a. By Theorem 6.1 on page 374, the ratio of the perimeters is 2 : 3. b. By Theorem 11.7 above, the ratio of the areas is 22 : 32, or 4 : 9.
11.3 Perimeter and Area of Similar Figures
737
★
EXAMPLE 2
Standardized Test Practice
You are installing the same carpet in a bedroom and den. The floors of the rooms are similar. The carpet for the bedroom costs $225. Carpet is sold by the square foot. How much does it cost to carpet the den?
USE ESTIMATION The cost for the den is 49 25
} times the cost for the
49 25
bedroom. Because } is a little less than 2, the cost for the den is a little less than twice $225. The only possible choice is D.
✓
A $115
B $161
C $315
D $441
Solution The ratio of a side length of the den to the corresponding side length of the bedroom is 14 : 10, or 7 : 5. So, the ratio of the areas is 72 : 52, or 49 : 25. This ratio is also the ratio of the carpeting costs. Let x be the cost for the den. 49 25
x 225
cost of carpet for den cost of carpet for bedroom
}5}
x 5 441
Solve for x.
c It costs $441 to carpet the den. The correct answer is D. A B C D
GUIDED PRACTICE
for Examples 1 and 2
1. The perimeter of n ABC is 16 feet, and its area is 64 feet. The perimeter
of n DEF is 12 feet. Given n ABC , n DEF, find the ratio of the area of n ABC to the area of nDEF. Then find the area of n DEF.
EXAMPLE 3
Use a ratio of areas
COOKING A large rectangular baking pan is 15 inches long and 10 inches wide. A smaller pan is similar to the large pan. The area of the smaller pan is 96 square inches. Find the width of the smaller pan. ANOTHER WAY For an alternative method for solving the problem in Example 3, turn to page 744 for the Problem Solving Workshop.
Solution 15 in.
First draw a diagram to represent the problem. Label dimensions and areas. Then use Theorem 11.7. If the area ratio is a 2 : b 2, then the length ratio is a : b.
10 in.
A 5 15(10) 5 150 in.2
?
A 5 96 in.2
Area of smaller pan 96 16 }} 5 } 5 } 150 25 Area of large pan
Write ratio of known areas. Then simplify.
Length in smaller pan 4 }} 5 } 5 Length in large pan
Find square root of area ratio.
4 c Any length in the smaller pan is } , or 0.8, of the corresponding length in the 5
large pan. So, the width of the smaller pan is 0.8(10 inches) 5 8 inches.
738
Chapter 11 Measuring Length and Area
REGULAR POLYGONS Consider two regular
polygons with the same number of sides. All of the angles are congruent. The lengths of all pairs of corresponding sides are in the same ratio. So, any two such polygons are similar. Also, any two circles are similar.
EXAMPLE 4
Solve a multi-step problem
GAZEBO The floor of the gazebo shown is
a regular octagon. Each side of the floor is 8 feet, and the area is about 309 square feet. You build a small model gazebo in the shape of a regular octagon. The perimeter of the floor of the model gazebo is 24 inches. Find the area of the floor of the model gazebo to the nearest tenth of a square inch. Solution All regular octagons are similar, so the floor of the model is similar to the floor of the full-sized gazebo. ANOTHER WAY In Step 1, instead of finding the perimeter of the full-sized and comparing perimeters, you can find the side length of the model and compare side lengths. 24 4 8 5 3, so the ratio of side lengths is 8 ft. 3 in.
96 in. 3 in.
32 1
STEP 1 Find the ratio of the lengths of the two floors by finding the ratio of the perimeters. Use the same units for both lengths in the ratio. 8(8 ft) Perimeter of full-sized 64 ft 64 ft 32 }} 5 } 5 } 5 } 5 } 1 24 in. 24 in. Perimeter of model 2 ft
So, the ratio of corresponding lengths (full-sized to model) is 32 : 1.
STEP 2 Calculate the area of the model gazebo’s floor. Let x be this area. (Length in full-sized)2
Area of full-sized Area of model
Theorem 11.7
309 ft2 x ft
Substitute.
5 }} }} 2 (Length in model)
} 5 } 5 }.
322 1
5} } 2 2 1024x 5 309
Cross Products Property
x ø 0.302 ft 2
Solve for x.
STEP 3 Convert the area to square inches. 144 in.2 1 ft
2 0.302 ft 2 p } 2 ø 43.5 in.
c The area of the floor of the model gazebo is about 43.5 square inches. (FPNFUSZ
✓
GUIDED PRACTICE
at classzone.com
for Examples 3 and 4
2. The ratio of the areas of two regular decagons is 20 : 36. What is the ratio
of their corresponding side lengths in simplest radical form? 3. Rectangles I and II are similar. The perimeter of Rectangle I is 66 inches.
Rectangle II is 35 feet long and 20 feet wide. Show the steps you would use to find the ratio of the areas and then find the area of Rectangle I.
11.3 Perimeter and Area of Similar Figures
739
11.3
EXERCISES
HOMEWORK KEY
5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 7, 17, and 27
★ 5 STANDARDIZED TEST PRACTICE Exs. 2, 12, 18, 28, 32, and 33
SKILL PRACTICE 1. VOCABULARY Sketch two similar triangles. Use your sketch to explain
what is meant by corresponding side lengths. 2.
EXAMPLES 1 and 2 on pp. 737–738 for Exs. 3–8
★ WRITING Two regular n-gons are similar. The ratio of their side lengths is is 3 : 4. Do you need to know the value of n to find the ratio of the perimeters or the ratio of the areas of the polygons? Explain.
FINDING RATIOS Copy and complete the table of ratios for similar polygons. Ratio of corresponding side lengths
Ratio of perimeters
Ratio of areas
3.
6 : 11
?
?
4.
?
20 : 36 5 ?
?
RATIOS AND AREAS Corresponding lengths in similar figures are given. Find
the ratios (red to blue) of the perimeters and areas. Find the unknown area. 5.
6.
A 5 240 cm2
15 cm A 5 2 ft 2
6 ft
20 cm
2 ft
7.
8.
A 5 210 in.2
A 5 40 yd2
5 yd 3 yd
9 in.
7 in.
EXAMPLE 3
FINDING LENGTH RATIOS The ratio of the areas of two similar figures is
on p. 738 for Exs. 9–15
given. Write the ratio of the lengths of corresponding sides. 9. Ratio of areas 5 49 : 16 12.
★
10. Ratio of areas 5 16: 121
11. Ratio of areas 5 121 : 144
MULTIPLE CHOICE The area of n LMN is 18 ft 2 and the area of n FGH is
24 ft 2. If n LMN , n FGH, what is the ratio of LM to FG? A 3:4
B 9 : 16
}
C Ï3 : 2
D 4:3
FINDING SIDE LENGTHS Use the given area to find XY.
13. n DEF , n XYZ D
4 cm
E
X
14. UVWXY , LMNPQ Y
A 5 198 in.2 W
A 5 88 in.2 N M
V F A 5 7 cm2
P X Z A 5 28 cm2
740
Chapter 11 Measuring Length and Area
U
L
10 in. P
Y
15. ERROR ANALYSIS In the diagram,
Rectangles DEFG and WXYZ are similar. The ratio of the area of DEFG to the area of WXYZ is 1 : 4. Describe and correct the error in finding ZY.
EXAMPLE 4
D G
W
X
Z
Y
E 12
F
ZY 5 4(12) 5 48
16. REGULAR PENTAGONS Regular pentagon QRSTU has a side length of
12 centimeters and an area of about 248 square centimeters. Regular pentagon VWXYZ has a perimeter of 140 centimeters. Find its area.
on p. 739 for Exs. 16–17
17. RHOMBUSES Rhombuses MNPQ and RSTU are similar. The area of RSTU
is 28 square feet. The diagonals of MNPQ are 25 feet long and 14 feet long. Find the area of MNPQ. Then use the ratio of the areas to find the lengths of the diagonals of RSTU. 18.
SHORT RESPONSE You enlarge the same figure three different ways. In each case, the enlarged figure is similar to the original. List the enlargements in order from smallest to largest. Explain.
★
Case 1 The side lengths of the original figure are multiplied by 3. Case 2 The perimeter of the original figure is multiplied by 4. Case 3 The area of the original figure is multiplied by 5. REASONING In Exercises 19 and 20, copy and complete the statement using always, sometimes, or never. Explain your reasoning.
19. Doubling the side length of a square ? doubles the area. 20. Two similar octagons ? have the same perimeter. 21. FINDING AREA The sides of n ABC are 4.5 feet, 7.5 feet, and 9 feet long.
The area is about 17 square feet. Explain how to use the area of n ABC to find the area of a n DEF with side lengths 6 feet, 10 feet, and 12 feet. 22. RECTANGLES Rectangles ABCD and DEFG are similar. The length of
ABCD is 24 feet and the perimeter is 84 square feet. The width of DEFG is 3 yards. Find the ratio of the area of ABCD to the area of DEFG. SIMILAR TRIANGLES Explain why the red and blue triangles are similar. Find the ratio (red to blue) of the areas of the triangles. Show your steps.
23. A 5 294 m2 D F
M
24.
T
E
U
Y 21 m
10 m
3 yd 308 L
N
X
V
W
25. CHALLENGE In the diagram shown, ABCD is a parallelogram.
A
The ratio of the area of n AGB to the area of n CGE is 9 : 25, CG 5 10, and GE 5 15.
B G
F
a. Find AG, GB, GF, and FE. Show your methods. b. Give two area ratios other than 9 : 25 or 25 : 9 for
pairs of similar triangles in the figure. Explain.
E
D
11.3 Perimeter and Area of Similar Figures
C
741
PROBLEM SOLVING 26. BANNER Two rectangular banners from this year’s music
festival are shown. Organizers of next year’s festival want to design a new banner that will be similar to the banner whose dimensions are given in the photograph. The length of the longest side of the new banner will be 5 feet. Find the area of the new banner.
3 ft
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
1 ft EXAMPLE 3
27. PATIO A new patio will be an irregular hexagon. The patio
will have two long parallel sides and an area of 360 square feet. The area of a similar shaped patio is 250 square feet, and its long parallel sides are 12.5 feet apart. What will be the corresponding distance on the new patio?
on p. 738 for Ex. 27
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
28.
MULTIPLE CHOICE You need 20 pounds of grass seed to plant grass inside the baseball diamond shown. About how many pounds do you need to plant grass inside the softball diamond?
★
A 6
B 9
C 13
D 20
60 ft
90 ft
softball diamond
baseball diamond
29. MULTI-STEP PROBLEM Use graph paper for parts (a) and (b). a. Draw a triangle and label its vertices. Find the area of the triangle. b. Mark and label the midpoints of each side of the triangle. Connect the
midpoints to form a smaller triangle. Show that the larger and smaller triangles are similar. Then use the fact that the triangles are similar to find the area of the smaller triangle. 30. JUSTIFYING THEOREM 11.7 Choose a type of polygon for which you know
the area formula. Use algebra and the area formula to prove Theorem 11.7 for that polygon. (Hint: Use the ratio for the corresponding side lengths in a two similar polygons to express each dimension in one polygon as } times b the corresponding dimension in the other polygon.)
31. MISLEADING GRAPHS A student wants to
show that the students in a science class prefer mysteries to science fiction books. Over a two month period, the students in the class read 50 mysteries, but only 25 science fiction books. The student makes a bar graph of these data. Explain why the graph is visually misleading. Show how the student could redraw the bar graph.
742
5 WORKED-OUT SOLUTIONS on p. WS1
Books Read Recently 60 50 40 30 20 10 0
★ 5 STANDARDIZED TEST PRACTICE
32.
★ OPEN-ENDED MATH The ratio of the areas of two similar polygons is 9 : 6. Draw two polygons that fit this description. Find the ratio of their perimeters. Then write the ratio in simplest radical form.
33.
★
11
EXTENDED RESPONSE Use the diagram shown at the right.
9
A
B
C
a. Name as many pairs of similar triangles as you can.
Explain your reasoning.
10
F D
b. Find the ratio of the areas for one pair of similar triangles. c. Show two ways to find the length of } DE.
E
34. CHALLENGE In the diagram, the solid figure is a cube. Quadrilateral
S
JKNM is on a plane that cuts through the cube, with JL 5 KL.
L K
R
a. Explain how you know that n JKL , n MNP. JK MN
J P
M
1 b. Suppose } 5 } . Find the ratio of the area of n JKL to the 3
T
P
area of one face of the cube.
N
c. Find the ratio of the area of n JKL to the area of pentagon JKQRS.
MIXED REVIEW PREVIEW Prepare for Lesson 11.4 in Exs. 35–38.
Find the circumference of the circle with the given radius r or diameter d. Use p ø 3.14. Round your answers to the nearest hundredth. (p. 49) 35. d 5 4 cm
36. d 5 10 ft
37. r 5 2.5 yd
38. r 5 3.1 m
Find the value of x. 39.
(p. 295)
40.
(p. 672)
41.
(p. 680)
x 10
1808
858
x8
888 x8
QUIZ for Lessons 11.1–11.3 1. The height of ~ ABCD is 3 times its base. Its area is 108 square feet. Find
the base and the height. (p. 720) Find the area of the figure. 2.
(p. 720)
13
3.
6.5
(p. 730)
4.
(p. 730)
5
12
8
4
6
4
3
5. The ratio of the lengths of corresponding sides of two similar heptagons
is 7 : 20. Find the ratio of their perimeters and their areas. (p. 737) 6. Triangles PQR and XYZ are similar. The area of n PQR is 1200 ft 2 and the
area of n XYZ is 48 ft 2. Given PQ 5 50 ft, find XY. (p. 737)
EXTRA PRACTICE for Lesson 11.3, p. 916
ONLINE QUIZ at classzone.com
743
Using
ALTERNATIVE METHODS
LESSON 11.3 Another Way to Solve Example 3, page 738 MULTIPLE REPRESENTATIONS In Example 3 on page 738, you used proportional reasoning to solve a problem about cooking. You can also solve the problem by using an area formula.
PROBLEM
COOKING A large rectangular baking pan is 15 inches long and 10 inches wide. A smaller pan is similar to the large pan. The area of the smaller pan is 96 square inches. Find the width of the smaller pan.
METHOD
Using a Formula You can use what you know about side lengths of similar
figures to find the width of the pan.
STEP 1 Use the given dimensions of the large pan to write expressions for the dimensions of the smaller pan. Let x represent the width of the smaller pan.
10 in.
A 5 96 in.2 x
15 in.
The length of the larger pan is 1.5 times its width. So, the length of the smaller pan is also 1.5 times its width, or 1.5x.
STEP 2 Use the formula for the area of a rectangle to write an equation. A 5 lw
Formula for area of a rectangle
96 5 1.5x p x 85x
Substitute 1.5x for l and x for w. Solve for a positive value of x.
c The width of the smaller pan is 8 inches.
P R AC T I C E 1. COOKING A third pan is similar to the
large pan shown above and has 1.44 times its area. Find the length of the third pan. 2. TRAPEZOIDS Trapezoid PQRS is similar
to trapezoid WXYZ. The area of WXYZ is 28 square units. Find WZ. P
R
9
X
4. REASONING n ABC , n DEF and the area
of n DEF is 11.25 square centimeters. Find DE and DF. Explain your reasoning. D
5 cm W
744
If another square has twice the area of the first square, what is its side length?
A
Y
6 P
3. SQUARES One square has sides of length s.
12
S
Chapter 11 Measuring Length and Area
Z
E B
8 cm
C
F
MIXED REVIEW of Problem Solving
STATE TEST PRACTICE
classzone.com
Lessons 11.1–11.3 1. MULTI-STEP PROBLEM The diagram below
4. SHORT RESPONSE What happens to the area
represents a rectangular flower bed. In the diagram, AG 5 9.5 feet and GE 5 15 feet.
of a rhombus if you double the length of each diagonal? if you triple the length of each diagonal? Explain what happens to the area of a rhombus if each diagonal is multiplied by the same number n. 5. MULTI-STEP PROBLEM The pool shown is a
a. Explain how you know that BDFH is a
rhombus.
right triangle with legs of length 40 feet and 41 feet. The path around the pool is 40 inches wide. R
b. Find the area of rectangle ACEG and the
U
P 40
S
area of rhombus BDFH. 41
c. You want to plant asters inside rhombus
BDFH and marigolds in the other parts of the flower bed. It costs about $.30 per square foot to plant marigolds and about $.40 per square foot to plant asters. How much will you spend on flowers? 2. OPEN-ENDED A polygon has an area of
48 square meters and a height of 8 meters. Draw three different triangles that fit this description and three different parallelograms. Explain your thinking. 3. EXTENDED RESPONSE You are tiling a 12 foot
by 21 foot rectangular floor. Prices are shown below for two sizes of square tiles.
18 in.
T Not drawn to scale
P
a. Find the area of nSTU. b. In the diagram, nPQR , nSTU, and the
scale factor of the two triangles is 1.3 : 1. Find the perimeter of nPQR. c. Find the area of nPQR. Then find the area
of the path around the pool. 6. GRIDDED ANSWER In trapezoid ABCD,
}i } AB CD, m∠ D 5 908, AD 5 5 inches, and CD 5 3 p AB. The area of trapezoid ABCD is 1250 square inches. Find the length (in CD. inches) of }
$2.25 $1.50
12 in.
7. EXTENDED RESPONSE In the diagram below,
n EFH is an isosceles right triangle, and n FGH is an equilateral triangle.
a. How many small tiles would you need for
F
the floor? How many large tiles? b. Find the cost of buying large tiles for the
floor and the cost of buying small tiles for the floor. Which tile should you use if you want to spend as little as possible? c. Compare the side lengths, the areas,